📂Matrix Algebra

The Spectrum and Decomposition Set of Matrices

Definition¹

The set $\sigma (A)$ of all eigenvalues of a square matrix $A$ is called the spectrum of $A$.

The complement set $\rho (A) = \mathbb{C} \setminus \sigma (A)$ of the spectrum is called the resolvent set of $A$.

Explanation

$$Ax = \lambda x$$

Consider the eigenvalue equation for matrix $A$. A vector $x$ that satisfies the above equation is called an eigenvector, and the constant $\lambda$ is called an eigenvalue. Modifying the equation slightly, we get the following. Here, $I$ is the identity matrix.

$$(A - \lambda I)x = 0$$

If $A - \lambda I$ is an invertible matrix, then we have $x = (A - \lambda I)^{-1}0 = 0$, and thus $x = 0$, making it not an eigenvector. Furthermore, $\lambda$ is also not an eigenvalue of $A$. Hence, the resolvent set of matrix $A$ is the set of $\lambda$s that make $A - \lambda I$ invertible.

$$\rho (A) = \left\{ \lambda : A - \lambda I \text { is invertible.} \right\}$$

Conversely, the spectrum of $A$ is the set of $\lambda$s that make $A - \lambda I$ non-invertible.

$$\sigma (A) = \left\{ \lambda : A - \lambda I \text { is singular.} \right\}$$

Etymology

The term spectrum originates from physics, where it refers to the division of light into its constituent wavelengths, displaying its colors, as learned in middle school through line spectra. Atoms emit specific energies (wavelengths) as they transition from an excited state to a ground state, known as the emission spectrum. The emission spectrum is unique to each element, representing its characteristics. Recall the flame test experiment with lithium, sodium, and potassium gas, each showing a unique flame color - this is the emission spectrum. The term “spectrum” of a matrix is named after the properties of light spectra, where each matrix can be considered an atom, and its eigenvalues are the unique wavelengths of energy it emits, collectively known as the spectrum.

This concept extends beyond physics and mathematics to describe “all possible values” more broadly. An example includes the “autism spectrum”, popularized by the recent trend in “Extraordinary Attorney Woo”. This term was coined because autism is not defined by a single symptom, but rather a range of characteristics.